Three factors inside the cube root is the same as one factor
outside the cube root.

There are now no variables which have 3 or more factors inside the
radical, but this is still not simplified because there is a
denominator under the radical. We must rationalize the denominator.
In order to bring out a factor of z in the bottom, there would need
to be 3 factors of z under the radical. We have 2 factors of z in the
bottom under the radical, so we need one more. We can multiply the
bottom by another factor of z so long as we multiply the top by the
same thing.

Three z's in the bottom under the radical is the same as one z outside the radical in the bottom.